Certainly changing all the earth straps and renewing the regulator/rectifier on my 900ss has made the world of difference on starting - as for a 5 year younger ST2 I'm unsure as to the electrical weaknesses.
I'd perhaps start with checking output voltages from the rectifier at different revs.
Do you have an alarm on the bike that could be failing in some way?
Good luck - Frank
PS - DSC member 'Derek' now has an ST4 and seems to be an electrical wizz, certainly helped me out with a few probs - might well be worth dropping a line:
http://www.ducatisportingclub.com/member.php?u=7468